Changes between Version 21 and Version 22 of u/johannjc/scratchpad5


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Timestamp:
10/07/15 10:27:17 (9 years ago)
Author:
Jonathan
Comment:

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  • u/johannjc/scratchpad5

    v21 v22  
    113113
    114114
    115 $T_0 + \displaystyle \sum_{\parallel, \perp} C \left [ \alpha_0 T^{\lambda+1}_0 +  \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda+1}_{\pm \hat{j}} +  \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T^{\lambda+1}_{\pm \hat{i} \pm \hat{j}} \right ] = $
    116 $T'_0 - \displaystyle \sum_{\parallel, \perp} D \left [  \alpha_0 T^{\lambda}_0T'_0 + \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda}_{\pm \hat{j}} T'_{\pm \hat{j}} + \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T^{\lambda}_{\pm \hat{i} \pm \hat{j}} T'_{\pm \hat{i} \pm \hat{j}} \right ]$
     115$T_0 + \Delta t \displaystyle \sum_{\parallel, \perp} C \left [ \alpha_0 T^{\lambda+1}_0 +  \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda+1}_{\pm \hat{j}} +  \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T^{\lambda+1}_{\pm \hat{i} \pm \hat{j}} \right ] = $
     116$T'_0 - \Delta t \displaystyle \sum_{\parallel, \perp} D \left [  \alpha_0 T^{\lambda}_0T'_0 + \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda}_{\pm \hat{j}} T'_{\pm \hat{j}} + \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T^{\lambda}_{\pm \hat{i} \pm \hat{j}} T'_{\pm \hat{i} \pm \hat{j}} \right ]$
    117117
    118118where
    119119
    120 $\alpha_0 = -\frac{2F_{ij}\delta_{ij} \Delta t}{\Delta x^2}$
    121 
    122 $\alpha_{\pm j} = \pm \frac{E_j \Delta t}{2 \Delta x} + \frac{F_{jj}\Delta t}{\Delta x^2}$
    123 
    124 $\alpha_{\pm i, \pm j} = \pm \pm \frac{F_{ij}\Delta t}{4 \Delta x^2}$
     120$\alpha_0 = -\frac{2F_{ij}\delta_{ij}}{\Delta x^2}$
     121
     122$\alpha_{\pm j} = \pm \frac{E_j }{2 \Delta x} + \frac{F_{jj}}{\Delta x^2}$
     123
     124$\alpha_{\pm i, \pm j} = \pm \pm \frac{F_{ij}}{4 \Delta x^2}$
    125125
    126126
    … …  
    137137|| $E_j$ || $A \left ( \partial_i  B_{ij} \right )$ ||
    138138|| $F_{ij}$ || $A B_{ij}$ ||
    139 ||$\alpha_0 $ ||$ -\frac{2F_{ij}\delta_{ij} \Delta t}{\Delta x^2}$ ||
    140 ||$\alpha_{\pm j}$ || $ \pm \frac{E_j \Delta t}{2 \Delta x} + \frac{F_{jj}\Delta t}{\Delta x^2}$ ||
    141 ||$\alpha_{\pm i, \pm j}$ || $ \pm \pm \frac{F_{ij}\Delta t}{4 \Delta x^2}$ ||
     139||$\alpha_0 $ ||$ -\frac{2F_{ij}\delta_{ij} }{\Delta x^2}$ ||
     140||$\alpha_{\pm j}$ || $ \pm \frac{E_j }{2 \Delta x} + \frac{F_{jj}\Delta t}{\Delta x^2}$ ||
     141||$\alpha_{\pm i, \pm j}$ || $ \pm \pm \frac{F_{ij}}{4 \Delta x^2}$ ||
    142142
    143143
    … …  
    175175|| $E_j$ || $A \left ( \partial_j \kappa \right )$ ||
    176176|| $F$ || $A \kappa$ ||
    177 ||$\alpha_0 $ ||$ -\frac{2F\delta_{ii} \Delta t}{\Delta x^2}$ ||
    178 ||$\alpha_{\pm j}$ || $ \pm \frac{E_j \Delta t}{2 \Delta x} +  \frac{F\Delta t}{\Delta x^2}$ ||
     177||$\alpha_0 $ ||$ -\frac{2F\delta_{ii} }{\Delta x^2}$ ||
     178||$\alpha_{\pm j}$ || $ \pm \frac{E_j }{2 \Delta x} +  \frac{F}{\Delta x^2}$ ||
    179179||$\alpha_{\pm i, \pm j}$ || $ 0$ ||
    180180
    … …  
    182182
    183183
     184=== Time stepping ===
     185The implicit method requires approximating
     186
     187$T*^{\lambda+1} = T^{\lambda + 1} +  \left ( \lambda + 1 \right ) T^{\lambda} \left [ \left ( T_{*} - T \right ) + \frac{\lambda}{2} \frac{T_{*} - T}{T} \right ]$
     188
     189but solving this accurately with a linear method requires
     190
     191$\frac{T_{*}-T}{T} << 1$
     192
     193but this restricts our time step.  Using the instantaneous derivatives, we can calculate the maximum time step as
     194
     195
     196$T_0'-T_0 \approx \displaystyle \sum_{\parallel, \perp} \left [ \alpha_0 T^{\lambda+1}_0 +  \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda+1}_{\pm \hat{j}} +  \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T^{\lambda+1}_{\pm \hat{i} \pm \hat{j}} \right ]$
     197
     198so
     199
     200$\Delta t < \epsilon \frac{T_0}{\displaystyle \sum_{\parallel, \perp} \left [ \alpha_0 T^{\lambda+1}_0 +  \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda+1}_{\pm \hat{j}} +  \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T^{\lambda+1}_{\pm \hat{i} \pm \hat{j}} \right ]}$
     201
    184202=== Explicit Case ===
    185203
    186204For the Explicit case, we just set $\phi = 0$ which sets $D = 0$ and $C=1$ and we have
    187205
    188 $T_0 +  \displaystyle \sum_{\parallel, \perp} \left [ \alpha_0 T^{\lambda+1}_0 +  \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda+1}_{\pm \hat{j}} +  \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T^{\lambda+1}_{\pm \hat{i} \pm \hat{j}} \right ] = T'_0$
    189 
     206$T_0 +  \Delta t \displaystyle \sum_{\parallel, \perp} \left [ \alpha_0 T^{\lambda+1}_0 +  \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda+1}_{\pm \hat{j}} +  \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T^{\lambda+1}_{\pm \hat{i} \pm \hat{j}} \right ] = T'_0$
     207
     208=== Explicit time stepping ===
     209
     210For the explicit scheme to be stable, we must have
     211
     212$\Delta t < \frac{1}{2\displaystyle \max_{\parallel, \perp, i} \left [ | \alpha_i T_i^{\lambda} |  \right ]}$
    190213
    191214